Quenched asymptotics for a 1-d stochastic heat equation driven by a rough spatial noise

In this note we consider the parabolic Anderson model in one dimension with time-independent fractional noise $dot{W}$ in space. We consider the case $H<frac{1}{2}$ and get existence and uniqueness of solution. In order to find the quenched asymptotics for the solution we consider its Feynman-Kac representation and explore the asymptotics of the principal eigenvalue for a random operator of the form $frac{1}{2} Delta + dot{W}$.